Balances and Abelian Complexity of a Certain Class of Infinite Ternary Words

Ondřej Turek

RAIRO - Theoretical Informatics and Applications (2010)

  • Volume: 44, Issue: 3, page 313-337
  • ISSN: 0988-3754

Abstract

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A word u defined over an alphabet 𝒜 is c-balanced (c∈ ℕ ) if for all pairs of factors v, w of u of the same length and for all letters a∈ 𝒜 , the difference between the number of letters a in v and w is less or equal to c. In this paper we consider a ternary alphabet 𝒜 = {L, S, M} and a class of substitutions ϕ p defined by ϕ p (L) = LpS, ϕ p (S) = M, ϕ p (M) = Lp–1S where p> 1. We prove that the fixed point of ϕ p , formally written as ϕ p ∞ (L), is 3-balanced and that its Abelian complexity is bounded above by the value 7, regardless of the value of p. We also show that both these bounds are optimal, i.e. they cannot be improved.

How to cite

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Turek, Ondřej. "Balances and Abelian Complexity of a Certain Class of Infinite Ternary Words." RAIRO - Theoretical Informatics and Applications 44.3 (2010): 313-337. <http://eudml.org/doc/250799>.

@article{Turek2010,
abstract = { A word u defined over an alphabet $\mathcal A$ is c-balanced (c∈$\mathbb N$) if for all pairs of factors v, w of u of the same length and for all letters a∈$\mathcal A$, the difference between the number of letters a in v and w is less or equal to c. In this paper we consider a ternary alphabet $\mathcal A$ = \{L, S, M\} and a class of substitutions $\varphi_p$ defined by $\varphi_p$(L) = LpS, $\varphi_p$(S) = M, $\varphi_p$(M) = Lp–1S where p> 1. We prove that the fixed point of $\varphi_p$, formally written as $\varphi_p^\infty$(L), is 3-balanced and that its Abelian complexity is bounded above by the value 7, regardless of the value of p. We also show that both these bounds are optimal, i.e. they cannot be improved. },
author = {Turek, Ondřej},
journal = {RAIRO - Theoretical Informatics and Applications},
keywords = {Balance property; Abelian complexity; substitution; ternary word; balance property},
language = {eng},
month = {10},
number = {3},
pages = {313-337},
publisher = {EDP Sciences},
title = {Balances and Abelian Complexity of a Certain Class of Infinite Ternary Words},
url = {http://eudml.org/doc/250799},
volume = {44},
year = {2010},
}

TY - JOUR
AU - Turek, Ondřej
TI - Balances and Abelian Complexity of a Certain Class of Infinite Ternary Words
JO - RAIRO - Theoretical Informatics and Applications
DA - 2010/10//
PB - EDP Sciences
VL - 44
IS - 3
SP - 313
EP - 337
AB - A word u defined over an alphabet $\mathcal A$ is c-balanced (c∈$\mathbb N$) if for all pairs of factors v, w of u of the same length and for all letters a∈$\mathcal A$, the difference between the number of letters a in v and w is less or equal to c. In this paper we consider a ternary alphabet $\mathcal A$ = {L, S, M} and a class of substitutions $\varphi_p$ defined by $\varphi_p$(L) = LpS, $\varphi_p$(S) = M, $\varphi_p$(M) = Lp–1S where p> 1. We prove that the fixed point of $\varphi_p$, formally written as $\varphi_p^\infty$(L), is 3-balanced and that its Abelian complexity is bounded above by the value 7, regardless of the value of p. We also show that both these bounds are optimal, i.e. they cannot be improved.
LA - eng
KW - Balance property; Abelian complexity; substitution; ternary word; balance property
UR - http://eudml.org/doc/250799
ER -

References

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